YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(b(x0))) → b(b(a(a(x0))))
b(a(b(x0))) → a(a(a(a(x0))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[a(x1)] =
0 -∞ 0
-∞ -∞ 0
-∞ 0 -∞
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b(x1)] =
0 1 0
-∞ 0 -∞
0 1 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
the rule
a(a(b(x0))) → b(b(a(a(x0))))
remains.

1.1 Bounds

The given TRS is match-bounded by 0. This is shown by the following automaton.